Dheyaa

Republic of Iraq Ministry of Higher Education And Scientific Research University of Baghdad College of Science

Microscopic Effective Charges and the

E2 Strength 1 1

( 2;2 0 )B E for even-

even Carbon Isotopes

A Thesis Submitted to Council of the College of Science

University of Baghdad in Partial Fulfillment of

The Requirements for the Degree of Master of Science

in Physics

By

Dheyaa Alwan AbdulHussain AL-Ibadi

B.Sc., University of Babylon (1998)

Prepared Under the Supervision of

Prof. Dr. Raad A. Radhi & Prof. Dr. Zaheda A. Dakhil

2013 1434

Certificate

We certify that preparation of this thesis, entitled Microscopic Effective

Charges and the E2 Strength 1 1

( 2;2 0 )B E for even-even Carbon

Isotopes was made under our supervision by Dheyaa Alwan

AbdulHussaen at the College of Science, University of Baghdad, in partial

fulfillment of the requirements for the degree of Master of Science in

Physics (Nuclear Physics).

Signature: Signature:

Name: Dr. Raad A. Radhi Name: Dr. Zaheda A. Dakhil

Title: Professor Title: Professor

Date: / / 2013 Date: / / 2013

In view of the available recommendations, I forward this thesis for

debate by the examining committee.

Signature:

Name: Dr. Raad M. S. Al-Haddad

Title: Professor

Head of Physics Department, College of Science

Date: / / 2013

We certify that we have read this thesis, entitled " Microscopic Effective

Charges and the E2 Strength 1 1

( 2;2 0 )B E for even-even Carbon

Isotopes'' and as examining committee, examined the student Dheyaa

Alwan AbdulHussaen on its contents, and that in our opinion it is

adequate for the partial fulfillment of the requirements for the degree of

Master of Science in Physics (Physics Nuclear).

Signature: Name: Dr. Abdul hussain A. Mahmood

Title: Professor (Chairman)

Date: / / 2013

Signature: Signature:

Name: Dr. Fadhil I. Sharrad Name: Dr. Gaith N. Flaiyh

Title: Assistant Professor Title: Assistant Professor

(Member) (Member)

Date: / / 2013 Date: / / 2013

Signature: Signature: Name: Dr. Raad A. Radhi Name: Dr. Zaheda A. Dakhil

Title: Professor Title: Professor (Supervisor) (Supervisor)

Date: / / 2013 Date: / / 2013

Approved by the University Committee of Postgraduate Studies.

Signature: Name: Dr. Saleh M. Ali

Title: Professor

Dean of the College of Sciences, University of Baghdad

Date: / / 2013

Dedication

To

My Wife

ENAS

Dheyaa

A

Acknowledgement

Praise be to Allah, Lord of the worlds, blessings and peace be upon his

messenger Mohammed and his household.

Firstly I would like to express my sincere and great appreciation to my

supervisors Prof. Dr. Raad A. Radhi and Prof. Dr. Zaheda A. Dakhil for

suggesting this project, appreciable directions, help and support throughout the

work.

I would like to thank the Dean of the college of science, and the head of

the department of physics for their support to perform this work.

I wish I have a word better than thanks to express my feelings to the

Ministry of Industrial represented by Minister Mr. AHMAD ALKURBULI,

and Al-Rasheed Co. Represented by the general director

Eng. JALAL H. HASSAN for their permission, help and support to get

this graduate degree.

I am grateful to the staff of the library of the college of Science, staff of

the central library of the University of Baghdad, and IVSL program for

providing references.

My thanks, gratitude and appreciation to all my friends and colleagues in

the M.Sc. program for their support and encouragement along the whole period

of the research work, especially to Dr. Fadhil I. Sharrad , Dr. Laith Ahmed

Najam and Dr. Fouad Al-Ajeeli to effective help for us to get some research

and sources used in this thesis.

B

Also special thanks for Dr. Arkan Refah, Mr. Ahmed, Mr. Bhaa, Mr.

Natheer, Mr. Malek, Mr.Moustfa and my closed friend and my colleagues in

this work Mr. Noory Sabah.

In this moment I can see your tears, I feel your happiness, I hear your

congratulations words, I am sad too much for losing you, thanks for all. To my

dear father, may God have mercy on you.

My deep and sincere thanks are to my dear mother for her support and

patience during my study.

Finally, to the light of my eyes Brothers and sisters the all faithful hearts

which help me.

Dheyaa

I

Contents

Contents I List of figures III

List of tables V

Abstract VI

Chapter One: Introduction

1.1 Introduction 1

1.2 Exotic Nuclei 5

1.3 Puzzle of Halo Nuclei 14

1.4 Literature Survey 19

1.5 Aim of the Present Work 34

Chapter Two: Theoretical Considerations

2.1 Electron Scattering 35

2.2 General Theory 36

2.3 The Reduced SingleParticle Matrix Elements of the Longitudinal Operator 39

2.4 The Many-Particles Matrix Elements 42

2.5 SingleParticle Matrix Element in SpinIsospin Formalism 42

2.6 The One-Body Density Matrix Elements (OBDM) 43

2.7 Corrections to the Form Factor 44

2.8 Electron Scattering Form Factor and Nucleon Density 46

2.9 Normalization of () 48 2.10 Root mean square radius in terms of occupation number 49

2.11 The Reduced Electromagnetic Transition Probability 51

2.12 The Relation between B (EJ) and B (EJ) 52

2.13 Core-polarization effects and effective charges 55

II

Chapter Three: Results, Discussion and conclusions

3.1 Introduction 59

3.2 The nucleus 10

C 63

3.3 The nucleus 12

C 65

3.4 The nucleus 14

C 66

3.5 The nucleus 16

C 67

3.6 The nucleus 18

C 69

3.7 The nucleus 20

C 70

3.8 Conclusions 73

3.9 Future work 75

Reference 76

III

List of Figures

CHAPTER ONE

Figure

1.1 The chart of nuclides. The valley of stability is indicated by the

black dots representing the stable nuclei in nature. The limits of

nuclear stability are indicated by the proton and neutron drip

lines, behind which no bound nuclei can exist. The double lines

indicate the magic numbers for the stable nuclei.

3

1.2 A chart of the nuclei in the light isotope region (adapted from

[4]). Unbound systems, being potentially populated through

one-proton removal reactions, form the outskirts of the neutron-

rich landscape.

4

1.3 Shows closed quantum systems (bound state) at left and Open

quantum systems (unbound state, nuclei far from stability) at the

right

5

1.4 The bound nuclear states come close to the continuum. 7

1.5 a) Stable nuclei, proton and neutron homogenously mixed, no

decoupling of proton and neutron distributions. (b) Unstable

nuclei, decoupling of proton neutron distributions (neutron-rich

system).

8

1.6 The size and granularity for the most studied halo nucleus 11

Li.

The matter distribution extends far out from the nucleus such

that the rms matter radius of 11

Li is large as 48

Ca, and the radius

of the halo neutrons large as for the outermost neutrons in 208

Pb

9

1.7 The behavior of density distributions halo nuclei. 10

1.8 Schematic view of the nuclear density as a function of distance

with the definition of the half density radius and the surface

thickness.

12

1.9 (a) Schematic view of 11

Li as 9Li core and two loosely bound

neutrons. (b) Visualization of the three-body system in 11

Li.

13

1.10 The Borromean structure in 11

Li. 13

IV

1.11 Shows the dependence of density distributions of a loosely

single proton and neutron (in a Woods-Saxon-type potential) on

binding energy (EB), orbital momentum quantum numbers (), and charge. The effect of the Coulomb and the centrifugal

potential are clear. The figure is taken from.

15

V

List of Tables

CHAPTER THREE

Table

3.1 The calculated root mean square matter radii of 1020

C compared

with the experimental data. 63

3.2 The calculated effective charges and B(E2) values of 10

C compared



C compared

with the experimental results. 65


C compared



C compared



C compared

with the experimental data.

70


C compared


3.8 The calculated B(E2) values using average effective chargesff ff

p n1.1 and 0.27e ee e e e , using psd MS for A=10,14 and spsdpf

(0+2) MS for A=16,18 and 20, compared with the experimental

data.

72

VI

Abstract

Quadrupole transitions and effective charges are calculated for even-even

Carbon isotopes (10 A 20) based on shell model with p and psd shell model

spaces for 10 A 14 and with large basis no core shell model with (0 2)

truncations for 16 A 20. Calculations with configuration mixing shell

model with these limited model spaces usually under estimate the measured E2

transition strength. Although the consideration of a large basis no core shell

model with (0 2) truncations for 16 A 20 where all major shells s, p,

sd and pf are used, calculations fail to describe the measured reduced transition

strength )02;2(11

EB without normalizing the matrix elements with effective

charges to compensate for the discarded space. Instead of using constant

effective charges, excitations out of major shell space are taken into account

through a microscopic theory which allows particlehole excitations from the

core and model space orbits to all higher orbits with 2 excitations, which are

called core-polarization effects. The two body Michigan sum of three ranges

Yukawa potential (M3Y) is used for the core-polarization matrix elements. The

simple harmonic oscillator potential is used to generate the single particle

matrix elements of all isotopes considered in this work. Two values of the size

parameter b are used, one for A=10-14 (group A) and the other one for A=16-

20 (group B), due to the difference in the root mean square (rms) matter radii

(Rm) between the two isotopes groups. The b value of each isotope is adjusted

to reproduce the experimental matter radius. The average value of b for group

A is 1.55 fm, while that of group B is 1.78 fm. These size parameters of the

harmonic oscillator almost reproduce all the rms matter radii for 10,12,14,16,18,20

C

isotopes within the experimental errors.

VII

Almost same effective charges are obtained for the neutron-rich C isotopes

which are smaller than the standard values. The calculated B(E2) values agree

very well with the experimentally observed trends of the recent experimental

data for the entire chain of even-even C isotopes.

The major contribution to the transition strength comes from the core-

polarization effects. The present calculations of the neutron-rich 16

C, 18

C and

20C isotopes show a deviation from the general trends of even-even nuclei in

accordance with experimental and other theoretical studies. The experimental

)02;2(11

EB values are very well reproduced confirming the anomalous

suppression in 16

C and 18

C. The configurations arises from the shell model

calculations with core-polarization effects which reproduce the experimental

B(E2) values, and give small effective charges confirm the formation of proton-

shell closure for 14,16,18

C. The average microscopic effective charges for

14,16,18,20C are 1.1e and 0.25e, for the proton and neutron, respectively which are

smaller than the standard empirical values for this mass region.

CHAPTER ONE

Introduction

CHAPTER ONE INTRODUCTION

1

CHAPTER ONE

1.1. Introduction

Many years ago beyond the exotic nuclei phenomena had been discovered,

this discovery is represented a split moment in the history of knowledge; it

was leading to a new era in the structure nucleus. In fact, this phenomenon

had been taken the attention of many physicists over the world with their

imaginations and efforts. This aid approached to understand the structures

and interpretations behavior of these nuclei. There have achieved in both

theoretical and experimental perspectives.

In the beginning, the few concepts should be understood due to exotic

nuclei. An atomic nucleus is a many-body Fermionic quantum system made

of neutrons and protons (called nucleons). Nuclear systems are extremely

small, with typical radii on the order of 10 -14

meters. Protons have a positive

electric charge and interact with one another through the Coulomb force,

which repels protons from one another and decreases in strength with 1/r2,

where r is the radial coordinate. Neutrons have no electric charge, and since

nuclear systems exist, there must be attractive forces stronger than the

Coulomb force at play on the length scale of nuclear existence. The

interaction between the protons and neutrons in the nucleus is called the

strong nuclear force: it is about 100 times stronger than the Coulomb force

on the length scale of a nucleus but is negligible for longer distances [1].

The properties of matter are determined by the number of protons and

neutrons in a nucleus. In particular, the number Z of protons characterizes

the different elements, from hydrogen to uranium; depending on the number

N of neutrons, each element can be present in nature in a variety of isotopes.

The stable elements that exist in nature are 92 and there are almost 300


2

stable isotopes. When displayed in the chart of the nuclides, see figure 1.1,

where the number of protons Z is plotted against the number of neutrons N,

the stable nuclei lie approximately along the diagonal from the lower left to

the upper right, called the valley of stability; this figure also is called

"nuclear landscape".

In nuclear landscape, one can classify it into a major three areas, the first

one is indicated by black dots represented the stable nuclei: i.e., infinite

lifetime, stable nuclei can be found around the so-called stability line, where

NZ. The second one is neutrons drip line (at the bottom of stable line that

shows in figure. 1.1, red line). The nucleus, in this line, has being known the

neutrons-rich. The last area, in figure 1.1, is proton drip line (above the

stable line that shows as blue line in the figure). The nucleus, in this line, has

been known the protons-rich. Figure 1.2 is a similar chart for nuclear

systems (light nuclides which have been concentrated in the present work).

In figure 1.1, the unstable area indicated in green color is called the Terra

Incognita, meaning in Latin unexplored land. At the upper right of

landscape, superheavy nuclei had been appeared, this discrimination

between neutrons-rich and protons-rich depending on N/Z ratio [2, 3]. The

professional scientists in this field called the area that lies beyond neutron-

rich and proton-rich edge of stability, where nuclei after edge of stability

become unbound. Figure 1.3 shows the nuclei in stability (bound state) and

at far from stability i.e. unbound state [1]. So the separation energy for

nucleons in this edge goes to zero.

Besides the search for the exact position of the drip lines in the landscape,

other motivations to investigate nuclei far from stability are the quests for

which combination of protons and neutrons can make up a nucleus.


3

Figure 1.1: The chart of nuclides. The valley of stability is indicated by the black dots

representing the stable nuclei in nature. The limits of nuclear stability are

indicated by the proton and neutron drip lines, behind which no bound nuclei

can exist. The double lines indicate the magic numbers for the stable nuclei.

These unstable isotopes lie away from the line of stability, in the region

within the neutron and proton drip lines, which defined previously, the limits

beyond which nuclei become unstable. Due to their distance from the valley

of stability, nuclei close to the drip lines are often referred to as "exotic

nuclei", indicating entities different from the most ordinary ones, available

in nature. The drip lines have experimentally been reached only for the

lightest 8 elements and their position in the nuclear chart is not exactly

known yet. Therefore, in order to test existing nuclear structure models, one

of the most important topics of research in nuclear physics is the exploration

of the nuclear chart to find out where the limits of nuclear binding energy.


4

One of the most remarkable results of these studies was the discovery of

novel nuclear structures in nuclei far from stability in the last decades of the

20th

century. In analogy with the luminous ring around the sun or the moon

seen under certain meteorological conditions, there are a type of nuclei was

referred to as "Halo Nuclei". Thus, the investigation of the properties of

exotic nuclei becomes one of the most important goals in nuclear physics

and it is now possible thanks by the development of modern technologies

available for radioactive beam production and heavy-ion accelerators [1].

Figure 1.2: A chart of the nuclei in the light isotope region (adapted from [4]). Unbound

systems, being potentially populated through one-proton removal reactions,

form the outskirts of the neutron-rich landscape.


5

Figure 1.3: Shows closed quantum systems (bound state) at left and open quantum

systems (unbound state, nuclei far from stability) at the right.

1.2. Exotic Nuclei

In the beginning, what is an exotic nucleus? The word 'exotic' is referred to

anything which is out of ordinary and attract a wide interest to realize it.

Theoretically, nuclei are finite many-body quantum systems which exhibit a

rich variety of single-particle, few-body, and many-body phenomena. In

nuclear physics, the exotic nuclei are phenomena for some light nuclei and it

has special conditions far from stability line or near drip-lines (neutron-

rich or proton-rich). In other words, exotic nuclei are nuclei with an

extraordinary ratio of protons and neutrons Z/N. Typically; they are very

unstable and decay into more stable nuclei, i.e. exotic nuclei so-called rare-

isotopes, which have a loosely binding energy, short-lived isotopes and large

isospin. Actually, exotic nuclei are available as secondary beams at many

radioactive beam facilities around the world [5]. These unstable nuclei are

generally weakly bound with few excited states. These exotic nuclei have a

thin cloud of nucleons orbiting at large distances from the others, forming

the core [5].


6

Neutron-rich nuclei have attracted much interest during the past decades

[6- 11], and this will continue to be so due to new generation radioactive

beam facilities in the world. These nuclei are characterized by a small

binding energy, and many new features originating from the weakly bound

property have been found, a halo and skin structures with a large spatial

extension of the density distribution [12], a narrow momentum distribution

[13].

Halo states, when approaching the drip lines the separation energy of the

last nucleon or pair of nucleons decreases gradually and the bound nuclear

states come close to the continuum, see figure 1.4. The combination of the

short range of the nuclear force and the low separation energy of the valence

nucleons results, in some cases, in considerable tunneling into the classical

forbidden region and a more or less pronounced halo may be formed. As a

result the spatial structure of the valence nucleons is very different from the

rest of the system and the valence and the core subsystems are to a large

extent separable [14]. Therefore, halo nuclei may be viewed as an inert core

surrounded by a low density halo of valence nucleon(s). They may therefore

be described in few-body or cluster models [10, 15]. The formation of halo

states is characteristic especially for light nuclei in the drip line regions,

although not all of these can form a halo. There is a large sensitivity of the

spatial structure and the separation energy close to the threshold. The

increase in size, which is due to quantum mechanical tunneling out from the

nuclear volume, will only take place if there is no significant Coulomb or

centrifugal barriers present. There are at present many well-established halo

states for light neutron-rich drip line nuclei consisting of a core plus one or

two neutrons.


7

Figure 1.4: The bound nuclear states come close to the continuum, where (proton) at left and

(neutron) at right.

Nuclei at the drip lines are led with nucleons up to the limit set by the

combination of the nuclear kinetic energy and the depth of the nuclear

potential well, resulting in a bound state close to the continuum. The binding

energy for this state is very small and, due to the short range of the nuclear

force, threshold effects appear in figure 1.5, (see also figure 1.3). The last

nucleons undergo the quantum-mechanical effect of tunneling, so that their

probability of being at large distances from the core is appreciable. This

referred to as 'halo' state for the first time in 1987 [8] and since then the term

halo has been referred to all exotic nuclei manifesting those properties. The

manifestation of the halo phenomenon is less evident if the halo nucleons are

in a state of large angular momentum >1, in which case the centrifugal

barrier lowers the probability of tunneling far from the core. Analogously,

the Coulomb barrier hinders the formation of proton halos, because the

repulsion between the protons and the nuclear core makes it difficult for the

nucleons to tunnel. The Coulomb repulsion also affects the absolute binding,

so that the proton drip line actually lies closer to the valley of stability than

the neutron drip line.


8

(a) (b)

Figuer1.5: (a) Stable nuclei, proton and neutron homogenously mixed, no decoupling of

proton and neutron distributions. (b) Unstable nuclei, decoupling of proton

neutron distributions (neutron-rich system).

A halo state consists of a veil of dilute nuclear matter that surrounds the

core. This is in contrast to the nuclear skin [16], which is essentially

difference in the radial extent of the proton and neutron distributions. A

loose definition of a halo would be that the halo nucleon(s) spend about 50%

of the time outside the range of the core potential and thus in the classically

forbidden region. The necessary conditions for the formation of a halo have

been investigated [17- 19] and it was found that, besides the condition of a

small binding energy for the valence particle(s), only states with small

relative angular momentum may form halo states. Two-body halos can thus

only occur for nucleons in s- or p-states.

The advent of rare-isotope beams has made it possible to explore how the

properties of nuclei evolve along a chain of isotopes extending to both

proton-rich and neutron-rich nuclei. This capability had been confirmed the

discovery of halo systems. Figure 1.6 gives a schematic illustration of the

sizes involved in the case of the two-neutron halo nucleus 11

Li. The binding

energy for the two halo neutrons is only about 300 keV [20] and they are

mainly in s- and p-states and can therefore tunnel far out from the core. It


9

turns out that the root mean square (rms) of 11

Li is similar to the radius of

48Ca while the two halo neutrons extend to a volume similar in size to

208Pb.

In addition to weak binding, the orbital occupied by the halo neutrons in Li

also contribute to its large size. In particular, the low angular momentum 2s

orbital intrudes into the p-shell. Such a reordering of orbital also leads to a

breakdown of the N = 8 shell gaps for Li.The neutron density distribution

in such loosely bound nuclei shows an extremely long tail, called the neutron

halo, see figure 1.7 [21].

Figure 1.6: The size and granularity for the most studied halo nucleus 11

Li. The matter

distribution extends far out from the nucleus such that the rms matter radius

of 11

Li is large as 48

Ca, and the radius of the halo neutrons large as for the

outermost neutrons in 208

Pb.


10

Figure 1.7: The behavior of density distributions halo nuclei.

Due to the confining Coulomb barrier which holds them closer to the core,

the formation of proton halo is rather difficult. Decoupling of core and

valence particles and their small separation energy are the important

criterion for a halo and in addition to these there is one another criterion that

the valence particle must be in a relatively low orbit angular momentum

state, preferable an s-wave, relative to the core, since higher l-values give

rise to a confining centrifugal barrier. The confining Coulomb barrier is the

reason for proton halos not so extended as neutron halos [22].

As a consequence the radius of such a nucleus is much larger than

expected. Although the density of a halo is very low, it strongly affects the

reaction cross section and leads to new properties in such nuclei, e.g. a very

narrow momentum distribution. This reflects the Heisenberg uncertainty

principle, when the distribution in coordinate space is wide, that in

momentum space is narrow.

2xx p (1-1)


11

In other word, the matter radius is much extended; there is a large

uncertainty in the position and thus a small uncertainty in the linear

momentum [22].

In the halo nucleus the three basic rules of nuclear density for nuclei close

to the valley of stability are broken [23]:

The first rule which is; the radius where the nuclear density is reduced to

half of the maximal density is expressed as 1 30 0

, 1.2R r A r fm is the

radius constant. The second one is protons and neutrons are homogenously

mixed in the nucleus, i.e. ( ) ( )p nr r , in halo nuclei this is only true for

the core. Finally, the distance from where the nuclear density drops from

90% of the maximal value to 10% of its maximal value, called the surface

thickness t, is constant and is about 2.3t fm, see figure 1.8. This feature is

true for nuclei in the valley of stability because of the nearly-constant

nucleon separation energy (6-8MeV) for stable nuclei. In general the surface

thickness or the surface diffuseness is expected to depend on the nucleon

separation energy. The neutron halo is the most pronounced case for small

separation energy, less than 1MeV [24].

Very heavier neutron-rich nuclei, rather than forming halos may instead

form neutron skin. The extension of the neutron skin can be expected to

have important influences on fusion cross sections because of the

localization of neutron matter at the surface of the nucleus. A complete

characterization of the neutron skin will require the knowledge of the single-

particle neutron states occupied in these nuclei in addition to their decay

properties and reaction rates. Further, the establishment of the thickness of

the skin determined by examining the difference between the nuclear matter


12

radius and the charge radius is a crucial quantity for testing models of

nuclear matter [23].

Figure 1.8: Schematic view of the nuclear density as a function of distance with the

definition of the half density radius R and the surface thickness t.

The most prominent and most studied example is 11

Li with a 9Li core and

two loosely bound neutrons see figure 1.9 early in 1975 Thibault et al. [24]

found very small two-neutron separation energy, but they did not attribute

this to a neutron halo. In 1985 Tanihata et al. [12] discovered the large

interaction cross section of 11

Li and attributed it to a halo property.

In the fragmentation of 11

Li, narrow momentum distributions were found

for the 9Li fragments, but not for the other fragments such as

8Li or

8He,

indicating that only the two neutrons in the last orbital contribute to the

formation of a neutron halo [23]. 11

Li was found to be a three-body system,

in which none of the internal two-body subsystems (dineutron and 10

Li) were

bound, giving it the name "Borromean halo nucleus" (This name is given

by the analogy of the three subsystems to the three rings of the coat of arms


13

of the Italian Borromean family) [8,10]. Figure 1.9 and figure 1.10 showed

three-body interactions are necessary for a full description of the nucleus.

Figure 1.9: (a) Schematic view of 11

Li as 9Li core and two loosely bound neutrons.

(b) Visualization of the three-body system in 11

Li.

Figure 1.10: The Borromean structure in 11

Li.


14

The obvious difference between a proton and a neutron is the Coulomb

interaction. A proton sees the Coulomb barrier at the surface of the nucleus.

Although the asymptotic form of the wave function may be the same, the

amplitude is damped because of the barrier. The height of the Coulomb

barrier exceeds 1 MeV, even if Z (atomic number) is as small as 5 (Boron),

if a normal Woods-Saxon-type density distribution is assumed. The other

type of the barrier that affects the wave function is the centrifugal barrier.

Since the centrifugal potential is proportional to (+1)/r2, its heights

depends heavily on the orbital angular momentum of the halo nucleon.

Figure 1.11 shows the density distributions of a loosely-bound single proton

and a single neutron in Woods-Saxon-type potential. Distributions are

shown separately for different orbitals: 2s, 1p and 1d. The effect of the

Coulomb interaction and the centrifugal potential is clearly visible. It is seen,

for example, for 2s, in which no centrifugal potential exists. The proton

density does not extend as much as that of neutrons. If one now compares

the neutron density tail for different , the tail is shorter for higher orbital

[23].

1.3. Puzzle of halo nuclei

In reference to halo stricter, the first hint came from the measurement of

the electric dipole transition between the bound states in 11

Be. Firstly, a

simple shell model picture of the structure of 11

Be suggested that its ground

state should consist of a single valence neutron occupying the 1p1/2 orbital

(the other six having filled the 1s1/2 and 1p3/2 orbital). But it was found that

the 2s1/2 orbital drops down below the 1p1/2 and this 'intruder' state is the

one occupied by the neutron, making it a (1/2) + ground state. The first


15

excited state of 11

Be, and the only other particle bound state, is the (1/2) -

state achieved when the valence neutron occupies the higher 1p1/2 orbital.

The very short lifetime for transition between these two bound states was

measured in 1983 [25] and corresponded to an E1 strength of 0.36 W.u

(Weisskopf Unit). It was found that this large strength could only be

understood if it is realistic single particle wave function is used to describe

the valence neutron in two states, which extended out to large distances due

to the weak binding.

Figure 1-11: The dependence of density distributions of a loosely single proton and

neutron (in a Woods-Saxon-type potential) on binding energy (EB), orbital momentum

quantum numbers (), and charge. The effect of the Coulomb and the centrifugal

potential are clear. The figure is taken from [23].


16

In the mid-1980's, the Berkeley experiment was carried out by Tanihata

and his group [12]. The interaction cross section of Helium (He) and

Lithium (Li) isotopes were measured and they found that the value was

much larger than the expected for the case of 6He and

11Li. These

corresponded to larger rms (matter density) than that is predicted by the

normal A1/3

dependence. After two years, Hansen and Jonson [8] proposed

that the large size of these nuclei is due to the halo effect. They explained the

large matter radius of 11

Li by treating it as a binary system of 9Li core plus a

di-neutron. Dineutron is a hypothetical point particle, implying the two

neutrons are stuck together i.e. the n-n system is unbound. Hence they

showed that the weak binding between this pair of clusters could form an

extended halo density.

High-energy proton elastic scattering is considered to be the best method

to obtain matter density distributions of nuclei. Proton elastic scattering

cross sections on various halo nuclei were measured at GSI Darmstadt using

liquid hydrogen target system [26, 27]. The matter density distributions had

been determined for 6,8

He and for 8,9,11

Li. The rms matter radii Rm = 2.45

0.10 fm for 6He and Rm = 2.53 0.08 fm for

8He were consistent with the

values obtained by the interaction cross sections [28].

The first value that had been measured on exotic nuclei using secondary

radioactive beam was the interaction cross section 1

( ) [29, 30]. It defined

as the total cross section for all processes in which projectile number of

nucleons is changed. From the interaction cross sections, one can define, the

interaction radius [29] using a simple geometric model:

21( , ) ( ( ) ( ))I IP T R P R T (1-2)


17

where RI (P) is radius of projectile and RI (T) is radius of target. It has been

shown that the interaction radius is more or less independent of the target.

Measuring the interaction cross section for different target one can obtain the

interaction radius of a projectile. In nuclear theory, it is known that stable

nucleus density is rather constant up to a certain radius from which it drops

to zero. The central density is quite similar from the lightest nuclei to the

heaviest. This led to the semiclassical liquid-drop model in which nuclei are

viewed as liquid droplet with a homogeneous density. In this model a

nucleus containing A nucleons is therefore seen as a sphere with a radius R

proportional to A1/3

[29].

13

oR r A with 1.2or fm (1-3)

If one assumes that the interaction radius is somehow a measure of the

nuclear size, it should follow the A1/3

law predicted from the liquid-drop

model. Some nuclei near the neutron drip line (like 6He,

8He,

11Li,

11Be and

14Be) exhibit anomalously high interaction radii in comparison with their

neighbors. This indicates that those nuclei have large nuclei radii due to an

extended matter density and /or a major deformation [30].

In the last two decades, development of experiments with a secondary

radioactive ion beam has extensively helped the studies on light neutron-rich

nuclei. One of the most striking features in neutron-rich nuclei is the nuclear

deformation. The deformation can be investigated experimentally and

theoretically, through their electromagnetic transitions. The general trend of

the 2+ excitation energy

1(2 )E

and the reduced electric quadrupole transition

strength between the first excited 2+ state and the 0

+ ground state,

1 1( 2;2 0 )B E

for even-even nuclei are expected to be inversely proportional

to one another [31]. However, recent experimental and theoretical studies of


18

the neutron-rich 16

C, 18

C and 20

C isotopes [32-35] show a deviation from the

general trends of even-even nuclei. A systematic study of transition rates for

these isotopes was recently conducted both theoretically and experimentally

[35-41].

The structure of light neutron-rich nuclei can be understood within the

shell model. Shell model within a restricted 1p model space was found to

provide good description for the 10

B natural parity level spectrum and

transverse form factors [42]. However, they were less successful for E2 form

factors and gave just 45% of the total observed E2 transition strength.

Expanding the model space to include 2 configurations in describing the

form factors, Cichocki et al. [42] had found that only 10% improvement was

realized. It had been long recognized [43] that these transitions have highly

collective properties. Those collective properties could be supplemented to

the usual shell model treatment by allowing excitations from the core and

model space orbits into higher orbits. The conventional approach to supply

that added ingredient to shell model wave functions is to redefine the

properties of the valence nucleons from those exhibited by actual nucleons

in free space to model-effective values [44]. Effective charges were

introduced for evaluating E2 transitions in shell-model studies to take into

account effects of model-space truncation. A systematic analysis had been

made for observed B(E2) values with shell-model wave functions using a

least-squares fit with two free parameters gave proton and neutron effective

charges, ffp

1.3e

e e and ff

n0.5

ee e [45] in sd-shell nuclei.

The role of the core and the truncated space can be taken into

consideration through a microscopic theory, which allows one particleone

hole (1p1h) excitations of the core and also of the model space to describe


19

these E2 excitations. These effects provide a more practical alternative for

calculating nuclear collectivity. These effects are essential in describing

transitions involving collective modes such as E2 transition between states

in the ground-state rotational band, such as in 18

O [46]. Umeya [39, 47]

calculated effective charges for quadrupole moments and transitions by

using first order perturbation. These theoretical results show that the

effective charges are smaller than the standard value in light- neutron rich

nuclei and imply decoupled quadrupole motions between protons and

neutrons in neutron-rich nuclei.

1.4. Literature Survey

Different kinds of theoretical models are currently being used to

investigate the structure of exotic nuclei. One of the early works was an

intermediate between shell model calculations and fully microscopic ones, is

the so called cluster-orbital shell model [48- 50].

When shell model calculations are considered, single particle wave

functions are provided from a certain mean field potential assuming stable

single particle motion. The shell model calculation is then carried out to treat

the residual interaction between nucleons moving on such stable single

particle orbits. In neutron rich nuclei where the last neutrons are forced to

occupy loosely bound or even unbound orbits of the mean potential, the

validity of shell model calculations can be questioned.

Tanihata et al. (1985) [12] calculated the interaction cross sections (I) for

all known Li isotopes 6Li-

11Li and

7Be,

9Be, and

10Be on targets Be, C, and

Al at 790 MeV/nucleon. Root mean square radii of these isotopes as well as

He isotopes had been deduced from the I by a Glauber-type calculation.

Appreciable differences of radii among isobars (6He-

6Li,

8He-

8Li, and

9Li-


20

9Be) had been observed for the first time. The nucleus

11Li showed a

remarkably large radius suggesting a large deformation or a long tail in the

matter distribution.

Tanihata et al. (1988) [51] measured 8B's interaction cross section. The

experiment found an interaction cross section very similar to those of the

neighboring Boron isotopes, and the deduced interaction radius followed the

A1/3

dependence typically seen in 'normal' nuclei. Those results for the 8B

nucleus were in contrast to measure interaction radii of nuclei found to have

neutron halos (which showed much larger than expected interaction radii).

That early result caused many to dismiss the possibility that 8B might

contain a proton halo.

Raman et al. [52] in 1988, completed a compilation of experimental results

for the electric quadrupole transition probability B(E2) between the 0+

ground state and the first excited 2+ state in even-even nuclei. For nuclei

away from closed shells, the SU(3) limit of the intermediate boson

approximation implies that the B(E2) values were proportional to

(epNp+enNn)2, where ep (en) is the proton (neutron) effective charge and Np

(Nn) refers to the number of valence protons (neutrons). The proportionality

was consistent with the observed behavior of B(E2) vs NpNn. For

deformed nuclei and the actinides, the B(E2) values calculated in a

schematic single-particle "SU(3)" simulation or large single-j simulation of

major shells successfully reproduced not only the empirical variation of the

B(E2) values but also the observed saturation of these values when plotted

against NpNn.

Minamisono et al. (1992) [53] measured the quadrupole moment of 8B and

used it as a probe of the structure. The quadrupole moment was found to be

|Q (8B) |= 68.3 2.1 mb, which was anomalously large, compared to shell


21

model predictions. The large value of the quadrupole moment was

interpreted by the authors as evidence for a proton halo within the 8B

nucleus.

Riisager et al. (1992) [17] calculated the radius of the two-body system for

different separation energies. The radius of the system became larger and

larger as separation energy decreased.

Otsuka et al. (1993) [54] proposed a variation shell model in order to

describe the structure of such nuclei. The model was applied to 11

Be, where

by using a Skyrme interaction the observed ground state of 11

Be nucleus was

reproduced correctly. In general mean field approximations proved to be of

restricted validity because of the weak binding of the halo neutrons. It was

realized that a more realistic approach to the halo structure would rely on

microscopic many-body model.

In 1994, Nakada and Otsuka [55] studied the E2 properties of A = 6-10

nuclei, including those of nuclei far from stability, which were studied by a

(0 2) shell-model calculation which includes E2 core-polarization

effects explicitly. The quadrupole moments and the E2 transition strengths in

A = 6-10 nuclei were described quite well by the present calculation. In that

paper, the result indicated that the relatively large value of the quadrupole

moment of 8B could be understood without introducing the proton halo in

8B. An interesting effect of the 2 core polarization was found for

effective charges used in the 0 shell model; although isoscalar effective

charges are almost constant as a function of nucleus, appreciable variations

are needed for isovector effective charges which play important roles in

nuclei with high isospin values.

Guimaraes et al. (1995) [56] had simultaneously measured the

longitudinal momentum distribution of 18

C, 17

C and 16

C following the


22

breakup of 19

C, 18

C and 17

C. In the 19

C data both the observation of narrow

momentum peak and a large cross section indicated the presence a one

neutron halo. The conclusion coincided with a shell-model calculation which

predicts an s state for the 19

C ground state. The width observed for the 18

C

one neutron breakup had been interpreted in the framework of fragmentation

model and suggested an increase of the absorptive radius cutoff possible due

to a higher neutron density on the nuclear surface. The 17

C data also showed

a narrow momentum peak, but it was wider and therefore incompatible with

the width deduced from the binding energy in the assumption of an s-state

neutron halo.

Experiments studying the momentum distributions of the breakup products

of 8B were also performed in order to study its structure. The breakup of

8B

into 7Be and a proton was studied, and the momentum distribution of the

7Be

products was measured. The first such experiment was done by Schwab et

al. (1995) [57] and the measured momentum distribution were found to be

very narrow when compared to those obtained from the breakup of 12

C and

16O. The deduced rms radius for the valence proton in

8B of 6.83 fm was

much larger than the expected for a normally bound proton, and was thought

to be a clear sign of a proton halo.

Bertulani and Sagawa (1995) [58] investigated the use of elastic and

inelastic scatterings with secondary beams of radioactive nuclei as a mean to

obtain information on ground state properties and transition matrix elements

to continuum states. In particular they discussed possible signatures of halo

wavefunctions in elastic and inelastic scattering experiments. The eikonal

approximation together with the t approximation (the double folding

approximation) yielded the simple and transparent formula. The model gave

very reasonable results for elastic scattering cross sections at forward angles


23

which could be used to test the ground state densities of radioactive nuclei.

The extended nuclear matter in exotic nuclei was manifest in the magnitude

of the elastic cross sections as well as in the position of the first minimum.

Pecina et al. (1995) [59] studied of quasi-elastic scattering of 8B from

12C

was performed with the goal of studying the structure of 8B. The measured

scattering cross sections were fitted using Optical Model potentials, and the

matter distribution determined. The deduced rms matter radius of 2.207 fm

for the 8B nucleus was small, and the authors interpreted this as evidence

against a substantial proton halo in 8B.

The studied of the momentum distributions of products from the breakup

of 8B was performed by Kelley et al. (1996) [60]. In that experiment, the

measured momentum distribution of the 7Be fragments was found to be

similar to that obtained by Schwab et al. (1995) [57] but after accounting for

absorption by the 7Be core, they deduced a rms radius for the valence proton

in 8B of 4.24 fm. Though the radius is larger than the systematic behavior of

nuclear radii (r0A1/3

=2.5 fm), it was much smaller than the rms halo radius

observed for the neutron halo nucleus 11

Li of 6.2 fm (1992) [61]. Those

results were interpreted as evidence against a large proton halo in 8B.

The first excited state of 17

Ne had been discussed by Chromik et al. (1997)

[62] were it populated via relativistic Coulomb excitation with a radioactive

beam of 17

Ne on a 197Au target and the subsequent -ray decay had been

observed. The 3

2

state was bound with respect to proton emission but

unbound to two-proton decay. The measured -ray yield accounts for

4314+19% of the predicted yield from an excitation cross section of 28 mb. It

was unlikely that the missing cross section could be attributed to two-proton


24

emission because the lifetime of this branch would have to be a factor 1700

smaller than predicted by standard barrier penetration calculations.

Kolata and Bazin in (2000) [63] had measured the longitudinal momentum

of the 13

B core fragment in one-neutron knockout from 14

B, on both 9Be and

197Au targets. The results of the experiment lend very strong support to the

idea that 14B is a neutron-halo system, the first odd-odd nucleus to display

this structure. It also appeared that, with the sole exception of 17

C at N=11,

all of the lowest-mass, particle-stable isotones from N=7-13 were halo

nuclei.

Zheng et al. [64] (2002) studied the reaction cross sections for 12,16

C had

been measured at the energy of 83 MeV by a new experimental method. The

larger enhancement of the 16

C reaction cross section at the low energy had

been used to study the density distribution of 16

C. The finite-range Glauber-

model calculations for different density distributions had been compared

with the experimental data. A large extension of the neutron density

distribution to a distance far from the center of the nucleus suggested the

formation of neutron halo in the 16

C nucleus.

Bazin et al. (2003) [65] used a new direct reaction: two-proton knockout

from neutron-rich nuclei, and they showed that a neutron-rich projectile

reacted with a light nuclear target, the knockout of two protons occurs as a

direct reaction. Consequently, the observed partial cross sections to

individual final levels conveyed selective information about nuclear

structure.

Zerguerras et al. (2004) [66] studied light proton-rich bound and unbound

nuclei by complete kinematics measurements which were produced by

means of stripping reactions of secondary beams of 20

Mg and 18

Ne. Their

work was the first measurement of its decay. As the decay scheme of the


25

nucleus could not be determined, two possible scenarios were proposed and

discussed. In addition, the decay of excited states in 17

Ne via two-proton

emission was observed.

In 2004, Imai et al. [32] presented a study of the electric quadrupole

transition from the first excited 2+ state to the ground 0

+ state in

16C was

studied through measurement of the lifetime by a recoil shadow method

applied to inelastically scattered radioactive 16

C nuclei. The measured mean

lifetime was 77 14(stat) 19 (sys.) ps. The central value of mean lifetime

corresponded to a B(E2; 21+ 0+) value of 0.63 e2fm4, or 0.26 Weisskopf

units. The transition strength was found to be anomalously small compared

to the empirically predicted value.

Sagawa et al. [36] in (2004) investigated static and dynamic quadrupole

moments of C and Ne isotopes by using the deformed Skyrme Hartree-Fock

model and also shell model wave functions with isospin-dependent core

polarization charges. It was shown at the same time that the quadrupole

moments Q and the magnetic moments of the odd C and odd Ne isotopes

depended clearly on assigned configurations, and their experimental data

will be useful to determine the deformations of the ground states of nuclei

near the neutron drip line. Electric quadrupole (E2) transitions in even C and

Ne isotopes were also studied using the polarization charges obtained by the

particle vibration coupling model with shell model wave functions.

Although the observed isotope dependence of the E2 transition strength was

reproduced properly in both C and Ne isotopes, the calculated strength

overestimates an extremely small observed value in 16

C.

Stanoiu et al. (2004) [67] studied the drip line nuclei through two-step

fragmentation and concluded that the neutron-rich 1720

C and 2224

O nuclei

had been performed by the in-beam -ray spectroscopy using the


26

fragmentation reactions of radioactive beams. The 2+ energy of

20C was

determined. Its low-energy value hinted for a major structural change at

N = 14 between C and O nuclei. Evidence for the non-existence of bound

excited states in either of the 23,24

O nuclei had been provided, pointing to a

large subshell effect at N = 16 in the O chain.

The electron scattering was studied on halo nuclei by Bertulani (2005)

[68], he was using the inelastic scattering of electrons on weakly-bound

nuclei to study with a simple model based on the long range behavior of the

bound state wavefunctions and on the effective-range expansion for the

continuum wavefunctions. It was shown that the cross sections for electro-

dissociation of weakly-bound nuclei reach ten milibarns for 10 MeV

electrons and increase logarithmically at higher energies.

Campbell et al. (2006) [69] established the measurement of excited states

in 40

Si and evidence for weakening of the N = 28 shell gap by detecting -rays

coincident with inelastic scattering and nucleon removal reactions on a

liquid hydrogen target. The large proton sub-shell gap at Z = 14 at means

that the evolution of excitation energies in the silicon isotopes was directly

related to the narrowing of the N = 28 shell gap.

Taqi and Radhi (2007) [70] had studied longitudinal form factors of the

low-lying, T=0, particle-hole states of 16

O, 12

C and 40

Ca using Random

Phase Approximation (RPA). The basis of single particle states was

considered to include 1s, 1p, 2s-1d and 1f-2p. The Hamiltonian was

diagonalized in the presence of Michigan three ranges Yukawa (M3Y)

interaction and compared with their previous results that depended on

Modified Surface Delta Interaction (MSDI). Admixture of higher

configuration up to 2p-1f was considered for the ground state. Effective

charges were used to account for the core polarization effect. Comparisons


27

were made to experimental data where the theoretical significance of the

calculations and its results were discussed.

In 2007, Hagino and Sagawa [37] applied a three-body model consisting of

two valence neutrons and the core nucleus 14

C in order to investigate the

ground state properties and electric quadrupole transition of the 16

C nucleus.

The calculated B(E2) value from the first 2+ state to the ground state showed

good agreement with the observed data with the core polarization charge

which reproduced the experimental B(E2) value for 15

C. It was also showed

that their calculations account well for the longitudinal momentum

distribution of the 15

C fragment from the breakup of the 16

C nucleus. They

pointed out that the dominant (d5/2)2 configurations in the ground state of

16C

played a crucial role in these agreements.

Radhi and Salman (2008) [46] had discussed Coulomb form factors for

collective E2 transitions in 18

O taking into account core polarization effects.

These effects are taken into account through a microscopic perturbation

theory including excitations from the core orbits and the model space

valence orbits to all higher allowed orbits with 10 excitations. The two-

body Michigan three range Yukawa (M3Y) interaction was used for the

core-polarization matrix elements. Two different model spaces with different

Hamiltonians were adopted. The calculations included the lowest four

excited 2+ states with excitation energies 1.98, 3.92, 5.25 and 8.21 MeV.

Ong et al. in 2008 [33] presented a studying of lifetime measurements of

first excited states in 16,18

C. In that article, the electric quadrupole transition

from the first excited 2+ state to the ground 0

+ states in

18C was studied

through a lifetime measurement by an upgraded recoil shadow method

applied to inelastically scattered radioactive 18

C nuclei. The measured mean

lifetime was 18.9 0.9(stat) 4.4(syst) ps, corresponding to a B(E2; 21+


28

0g.s.+ ) value of 4.3 0.2 1.0 e

2fm

4, or about 1.5 Weisskopf units. The mean

lifetime of the first excited 2+ state in

16C was re-measured to be 18.3 1.4

4.8 ps, about four times shorter than the value reported previously. The

discrepancy between the two results was explained by incorporating the -

ray angular distribution measured in that work into the previous

measurement. The transition strengths were hindered compared to the

empirical transition strengths, indicating that the anomalous hindrance

observed in 16

C persists in 18

C.

Wiedeking et al. in 2008 [38] studied the lifetime of the 21+ state in

16C

which had been measured with the recoil distance method using the 9Be

(9Be, 2p) fusion-evaporation reaction at a beam energy of 40 MeV. The

mean lifetime was measured to be 11.7(20) ps corresponding to a B (E2;

21+0+) value of 4.15(73) e2fm4, consistent with other even-even closed

shell nuclei. Their result did not support an interpretation for decoupled

valence neutrons.

Radhi et al. (2008) [71] calculated large basis no core shell model to study

the elastic and inelastic electron scattering on 19

F. All major shells s, p, sd

and pf were considered with (0 + 2) truncations. Excitations out of major

shell space were taken into account through a microscopic theory that allows

particlehole excitations from the sd and pf shell orbits to all higher orbits

with 2 excitations. Excitations out the no core shell model space were

essential in obtaining a reasonable description of the longitudinal and

transverse electron scattering form factors.

Chatterjee et al. (2008) [72] investigated the role of higher multipole

excitations in the electromagnetic dissociation of one-neutron halo nuclei

within two different theoretical models were finite-range distorted-wave

Born approximation and another in a more analytical method with a finite-


29

range potential. They also showed within a simple picture, how the presence

of a weakly bound state affected the breakup cross-section.

Radhi (2009) [73] investigated Coulomb excitations of open sd-shell

nuclei. Microscopic theory was employed to calculate the C2 form factors

for the first two excited 2+ states in

22Ne,

26Mg and

30Si. Those collective

transitions were discussed, taking into account core-polarization effects.

Remarkable agreements were obtained between the measured and calculated

form factors for the first excited 2+ state. No strong conclusion could be

drawn for the second excited 2+ state.

Radhi et al. (2009) [74] analyzed elastic and inelastic electron scattering

from 9Be using large-basis shell model calculations. All major shells s, p, sd

and pf were considered. The calculations were performed with truncated

space, with configurations up to 2. Such calculations fail to describe the

electron scattering data without normalizing the matrix elements with

effective charges. Instead of using constant effective charges, one particle

one hole excitations were taken into account from all the major shell orbits

into all higher allowed orbits with excitations up to 10. Excitations up to

6 seemed to be large enough for sufficient convergence. Those

excitations were essential in obtaining a reasonable description of the data.

Calculations were presented for the transitions from JT = 3/2

1/2 to

JT = 3/2

1/2, 5/2

1/2 and 7/2

1/2.

Umeya et al. (2009) [39] presented the theoretical approach based on the

shell model to calculate effective charges of electric quadrupole transitions.

The shell-model calculation with single-particle 2 excitations in the first

order perturbation qualitatively reproduced existing experimental B(E2)

values for carbon isotopes with neutron number 5N16 and showed a

sudden change of the isovector effective charge beyond N = 8.


30

The calculations of the neutron skin and its effect in atomic parity

violation adopted by Brown et al. (2009) [75] to atomic parity non

conservation (PNC) in many isotopes of Cs, Ba, Sm, Dy, Yb, Tl, Pb, Bi, Fr,

and Ra. Three problems were addressed: (I) neutron-skin-induced errors to

single-isotope PNC, (II) the possibility of measuring neutron skin using

atomic PNC, and (III) neutron-skin-induced errors for ratios of PNC effects

in different isotopes. In the latter case the correlations in the neutron skin

values for different isotopes lead to cancellations of the errors; this makes

the isotopic ratio method a competitive tool in a search for new physics

beyond the standard model.

Coraggio et al. (2010) [76] studied neutron-rich carbon isotopes in terms

of the shell model employing a realistic effective Hamiltonian derived from

the chiral N3LOW nucleon-nucleon potential. The single-particle energies

and effective two-body interaction had been both determined within the

framework of the time-dependent degenerate linked-diagram perturbation

theory. The calculated results were in very good agreement with the

available experimental data, providing a sound description of that isotopic

chain toward the neutron dripline. The correct location of the drip line was

reproduced.

Wuosmaa et al. [77] in 2010, studied the 15

C (d, p)16

C reaction in inverse

kinematics using the Helical Orbit Spectrometer at Argonne National

Laboratory. Neutron-adding spectroscopic factors gave a different probe of

the wave functions of the relevant states in 16

C. Shell-model calculations

reproduced both the present transfer data and the previously measured

transition rates.

Hagino et al. (2011) [78] proposed a simple schematic model for two-

neutron halo nuclei. In that model, the two valence neutrons moved in a one-


31

dimensional mean field, interacting with each other via a density-dependent

contact interaction. They investigated the ground state properties, and

demonstrate that the dineutron correlation could be realized with that simple

model due to the admixture of even- and odd-parity single-particle states.

They then solved the time-dependent two-particle Schrdinger equation

under the influence of a time-dependent one-body external field, in order to

discuss the effect of dineutron correlation on nuclear breakup processes. The

time evolution of two-particle density showed that the dineutron correlation

enhanced the total breakup probability, especially for the two-neutron

breakup process, in which both the valence neutrons were promoted to

continuum scattering states. They found that the interaction between the two

particles definitely favors a spatial correlation of the two outgoing particles,

which were mainly emitted in the same direction. Neutron halo deformed

nuclei from a relativistic Hartree-Bogoliubov model in a Woods-Saxon

basis.

Hammer and Phillips (2011) [79] computed E1 transitions and electric

radii in the Beryllium-11 nucleus using an effective field theory (EFT) that

exploited the separation of scales in that halo system. They fixed the

leading-order parameters of the EFT from measured data on the 1/2+ and

1/2 levels in

11Be and the B(E1) strength for the transition between them.

Also they obtained predictions for the B(E1) strength for Coulomb

dissociation of the 11

Be nucleus to the continuum. They computed the charge

radii of the 1/2+ and 1/2

states. Agreement with experiment within the

expected accuracy of a leading-order computation in the EFT was obtained.

That paper was discussed how next-to-leading-order (NLO) corrections

involving both s-wave and p-wave 10

Beneutron interactions affected their

results, and displayed the NLO predictions for quantities which were free of


32

additional short-distance operators at that order. Information on neutron 10

Be

scattering in the relevant channels was inferred.

Zhou et al. (2011) [80] studied and discussed the halo phenomenon in

deformed nuclei by using a fully self-consistent deformed relativistic

Hartree-Bogoliubov model in a spherical Woods-Saxon based with the

proper asymptotic behavior at large distance from the nuclear center. Taking

a deformed neutron-rich and weakly bound nucleus 44

Mg as an example and

by examining contributions of the halo, deformation effects, and large spatial

extensions, they showed a decoupling of the halo orbital's from the

deformation of the core.

Petri et al. (2011) [40] reported the first measurement of the lifetime of the

21+ state was in the near-dripline nucleus

20C. The deduced value of 21+ =

9.82.8()1.1+0.5 (syst) ps gave a reduced transition probability of B(E2;

21+0g.s.

+ ) = 7.51.7+3.0 ()0.4

+1.0 (syst) e2fm

4 which was in a good agreement

with a shell model calculation using isospin-dependent effective charges.

In 2012, Voss et al. [35], presented the lifetime of the first excited 2+ state

which was measured with the Kln/NSCL plunger via the recoil distance

method to be (21+) = 22.4 0.9()2.2

+3.3 (syst) ps, which corresponds to a

reduced quadrupole transition strength of B(E2; 21+ 01

+) = 3.640.14+0.15

()0.47+0.40 (syst) e

2fm

4. In addition, an upper limit on the lifetime of a

higher-lying state feeding the 21+ state was measured to be < 4.6 ps. The

results were compared to the large-scale ab initio no-core shell model

calculations using two accurate nucleon-nucleon interactions and the

importance-truncation scheme. That comparison provided strong evidence

that the inclusion of three-body forces was needed to describe the low-lying

excited-state properties of that A = 18 system.


33

Nakamura (2012) [81] discussed how the neutron halo nuclei were studied

by the breakup reactions at relativistic energies. In the Coulomb breakup of

halo nuclei, enhancement of the electric dipole strength at low excitation

energies (soft E1 excitation) was observed as a unique property for halo

nuclei. The mechanism of the soft E1 excitation and its spectroscopic

significance was shown as well as the applications of the Coulomb breakup

to the very neutron rich 22

C and 31

Ne, which was measured at 230-240

MeV/nucleon at the new-generation RI beam facility, RIBF (RI Beam

Factory), at RIKEN. Evidence of halo structures for those nuclei was

provided as enhancement of the inclusive Coulomb breakup, which was a

useful tool for the low-intense secondary beam. He also showed that the

breakup with a light target (C target), where nuclear breakup was a dominant

process, could be used to extract the spectroscopic information of the

removed neutron. The combinatorial analysis was found very useful to

extract more-detailed information such as the spectroscopic factor and the

separation energy. Prospects of the breakup reactions on neutron-drip line

nuclei at RIBF at RIKEN were also briefly presented.

In 2012, McCutchan et al. [41] studied lifetime of the 21+ state in

10C. It

was measured using the Doppler shift attenuation method. That

measurement, combined with that determined for 10

Be 9.2(3) e2fm

4 testing

the structure of those states, including the isospin symmetry of the wave

functions. By adopting Quantum Monte Carlo calculations, the reproduced

the 10Be B(E2) value by using realistic two- and three-nucleon

Hamiltonians can predict a larger 10C B(E2) probability. In these

calculations, the sensitivity to the admixture of different spatial symmetry

components in the wave functions and to the three-nucleon potential is

considered.


34

1.5. Aim of the present work

The aim of the present work is using the fundamental relations to get the

reduced transition strength B(E2) from the first-excited 2+ state to the ground

state for some even-even carbon isotopes in the range A=10-20. Also, a

good imagination for the nuclear structure of these isotopes has adopted

using different model spaces and interactions. These B(E2) values represent

basic nuclear information complementary to the knowledgement of the

energies of low-lying levels in these nuclides. As well, the calculations are

depending on basic equations which explained the relation between different

parameters to reproduce the rms to fix the size parameter b of the single-

particle (HO) wave function. In this study, the size parameter plays the role

of a characteristic length of the harmonic-oscillator potential. Also, the rms

radius matter for these isotopes are reproduced and compared with the

available experimental data depending on the size parameters b for both halo

and core. For the quadrupole transition strength, excitation from the core and

model space will be taken into consideration through first-order perturbation

theory, where 1p-1h excitations are taken into considerations. These 1p-1h

excitations from the core and model space orbital are considered into all

higher allowed orbits with 2 excitations. Effective charges are calculated

in this work through the microscopic theory discussed above for the different

model spaces used, and compared with each of the stable nuclei and the

standard effective charges ( 1.3 , 0.5 )eff effp ne e e e . The calculated B(E2)

values will be compared with the most recent experimental data. All

calculations presented in this work are performed by a Fortran 90 computer

program written by Prof. Dr. R. A. Radhi.

Theoretical Considerations

CHAPTER Two

CHAPTER TWO THEORETICAL CONSIDERATIONS

35

CHAPTER TWO

Theoretical Considerations

2.1. Electron scattering

Electron scattering method is an excellent tool for studying nuclear

structure because there are two reasons; one of them is that the interaction is

known, as the electron interacts electromagnetically with the local charge

and current density in the target. Since this interaction is relatively weak,

one can make measurement without greatly disturbing the structure of the

target. The second one is the advantage of electrons for fixed energy loss to

the target, one can vary the three-momentum transfer q and map out the

Fourier transforms of the static and transition densities [82].With electron

scattering one can immediately relate the cross section to the transition

matrix elements of the local charge and current density operators and this is

directly to the structure of the target itself.

The scattering of electrons from a target nucleus can be distinguished two

ways. In one, the nucleus is left in its ground state after the scattering and the

energy of the electrons is unchanged which is known as "Elastic electron

scattering". In the other, the scattered electron leaves the nucleus in

different excited state and has a final energy reduced from the initial just by

the amount taken up by the nucleus in its excited state, it is called "Inelastic

electron scattering" [83-85].

Electron scattering process can be explained according to the first Born

approximation as an exchange of virtual photon, which carrying a

momentum between the electron and nucleus [86]. The first Born

Approximation is being valid only if 1 , where is the fine

structure constant. According to this approximation two types of electron


36

scattering from the nucleus are recognized. The first is the longitudinal or

Coulomb scattering .CoJF in which the electron interacts with the charge

distribution of the nucleus where the interaction is considered as an

exchange of a virtual photon carries a zero angular momentum along the

direction of the momentum transfer q . This process gives all information

about the nuclear charge distribution. In the second type, the electron

interacts with the magnetization and current distributions. This process is

considered as an exchange of a virtual photon with angular momentum +1

along q direction. This type of scattering is called transverse scattering TJF

and it provides the information about the nuclear current and magnetization

distributions. The transverse form factor can be divided into two kinds;

transverse electric and magnetic form factors according to the parity

selection rules.

2.2. General Theory

In the plane-wave Born approximation (PWBA), the differential cross-

section for the scattering of an electron from a nucleus of charge (Ze) and

mass (M) into a solid angle (d) is given by [86, 87]:

2

( , )rec JJMott

d df F q

d d

(2-1)

where Mott

d

d

is the Mott cross-section for high-energy electron from a

point spineless nucleus, which is given by [88]:

2

2

cos( 2)

2 sin ( 2)Motti

Zd

d E

(2-2)


37

where 2 1 / 137e c

is the fine structure constant, is the scattering

angle and Ei is the energy of incident electron.

The recoil factor of the nucleus is given by [88]:

1

221 sin ( 2)irecE

fM

(2-3)

The total nuclear form factor for elastic scattering or for inelastic scattering

between an initial state i and final state f is ,( )JF q of a given multipolarity

J, which is a function of momentum transfer q and the angle of scattering ,

contains two parts, longitudinal (Coulomb) part L

JF and transverse (electric

and magnetic) part T

JF which can be written as [86]:

4 22 2

2 22

( , ) ( ) tan ( 2) ( )2

L TJ J J

q qF q F q F q

q q

(2-4)

The four-momentum transfer q is given by, (with =c=1):

2 2 2( )i f

q q E E (2-5)

where,

2 2 24 sin ( 2) ( )i f i f

q E E E E (2-6)

2 2 24 sin ( 2)i f

q E E

(2-7)

where i f

E E , i

E and f

E are the initial and final total energies of the

incident and scattered electron, respectively. The squared transverse form

factor can be expressed as the sum of the squared electric form factor and

squared magnetic form factor as follows:

22 2..( ) ( ) ( )magT ElJ J JF q F q F q (2-8)


38

The form factor of multipolarity as a function of momentum transfer is

written in terms of the reduced matrix elements of the transition operator

( )JT q as [86]:

22

2

4( ) ( )

(2 1)J Jf i

i

F q J T q JZ J

(2-9)

The symbol represents longitudinal L or transverse T (electric El or

magnetic mag.), Ji and Jf are the total angular momentum of the initial and

the final states, respectively. The JT )(q is the electron scattering multipole

operator, and is the reduced matrix element.

The nuclear states have a well-defined isospin. So using the Wigner-Eckart

theorem in isospin space, the form factor can be written in terms of the

matrix element reduced both in total angular momentum (spin) (J) and

isospin (T) (triple-bar matrix elements) [87].

2

2

4( )

(2 1)J

i

F qZ J

2

0,1

( 1)f z f

f i

T Tf i

f f J T i iT z zT

T T TJ T T J T

T M T

. (2-10)

where,

2Z

Z NT

(2-11)

The bracket . . .

. . .

denotes the 3j-symbols. Since f iz z

T T for electron

scattering, then MT=0.

The multipolarity J in equation (2-10) is restricted by the angular

momentum selection rule:


39

i f i f

J J J J J (2-12)

The parity selection rules [89]:

( 1)El J (2-13)

. 1( 1)mag J (2-14)

2.3. The Reduced SingleParticle Matrix Elements of the

Longitudinal Operator

The longitudinal scattering comes from the interaction of the electrons

with the charge distribution of the nucleus. The longitudinal operator is

defined as [86]:

( ) ( ) ( ) ( , )co r zJ M J JMT q d r j qr Y r t (2-15)

where,

( )Jj qr is the spherical Bessel function.

( )rJMY is the spherical harmonics function.

( , )zr t is the nucleon charge density operator which is given by:

1

( , ) ( ) ( )z

z zi ii

r t e t r r

(2-16)

Here, the sum is over protons,

1 ( )( ) , 2

2z

z z zi

e t t

and ( )i

r r is Dirac delta function.

From equations (2.15) and (2.16), the longitudinal operator becomes:

( ) ( ) ( ) ( )co z rJ M J JMT q e t j qr Y (2-17)

The reduced single-particle matrix element of the longitudinal operator

between the final state and the initial state can be written as:


40

.1 1 ( ) ( ) ( )2 2z

Coz rJt J JT e t n l j qr n l l j Y l j

(2-18)

The reduced matrix element of spherical harmonic is given by [88]:

121 1 1( ) ( 1) 1 ( 1)

2 2 2

j l l JrJl j Y l j

12(2 1)(2 1)(2 1)

1 14 02 2

j J jj j J

(2-19)

Equation (2.18) can be written as:

, ( ) ( , ) ( , )

ZJ t Z J JT e t P l l C j j

( )J

n l j qr n l

(2-20)

where JP and JC are the coefficients of electric parity-selection rules, which

given by [84]:

1

( , ) 1 ( 1)2

l l J

JP l l

(2-21)

11 22

(2 1)(2 1)(2 1)( , ) ( 1)

4

1 10

2 2

j

J

j j JC j j

j J j

(2-22)

The radial parts ( )nln l R r are normalized as:

0

2 2( ) 1nl

R r r dr

(2-23)


41

where,

0

2( ) ( ) ( ) ( )J J n l n ln l j qr n l dr r j qr R r R r

(2-24)

By using the harmonic oscillator potential with the size parameter b, the

radial matrix elements of Bessel function can be solved analytically as [86]:

1222( ) exp( ) ( 1) ! ( 1)!

(2 1)!!

JJ

Jn l j qr n l y y n nJ

121 1

( ) ( )2 2

n l n l

1 1

0 0

( 1)

! !( 1)!( 1)!

m mn n

m m m m n m n m

1( ( 2 2 3))2

3 3( ) ( )

2 2

l l m m J

m l m l

1 3

( 2 2 ); ;2 2

F J l l m m J y

(2-25)

where

2

2

bqy

, is gamma function and F is the confluent

hypergrometric function which may be evaluated using [86, 90]:

2( 1)

( , , ) 1( 1)2!

A A A yF A B y y

B B B

(2-26)

where A and B are positive integers.


42

2.4. The Many-Particles Matrix Elements

The many-particles reduced matrix elements of ,

z

J tT , operator can be

expressed as the sum of the product of the elements of the one-body density

matrix (OBDM) times the single-particle matrix elements [84]:

,

,

||| ||| ( , , , ) ||| |||J T

J T J Tf iJ T J OBDM i f j j T

(2-27)

where the reduced single-particle matrix element a ||| |||J T

T is given

in equation (2.20).

For inelastic scattering, the sum extends over all pairs of single-particle

states in the model space, but elastic longitudinal scattering, the sum

including the core orbits.

2.5. SingleParticle Matrix Element in SpinIsospin Formalism

The operator ,

zJ t

T can be written in terms of the isoscaler part and the

isovector part. Using Wickner-Eckart theorem, the single-particle matrix

element reduced in spin space can be written in terms of that reduced in

spin-isospin space as follows:-

12

, , 0

1 101 1 1 1 , , ( 1) , ,2 2

2 2 2 20

z

z

t

z zJ t J T

z z

j t T j t j T j

t t

12

, 1

1 11 1 1( 1) , ,2 2

2 20

z

z z

t

J Tj T j

t t

(2-28)


43

The reduced single-particle matrix element in spin-isospin space, given in

equation (2-27) becomes:

2 1 ||| ||| ( ) || ||

2 zzzJ T T J tt

TT I t T

(2-29)

where,

= 1 = 0

(1)1

2 = 1

(2-30)

The single-particle matrix element || ||z

J tT can be calculated

according to equation (2-18).

2.6. The One-Body Density Matrix Elements (OBDM)

In isospin representation, the value of JTOBDM is obtained from the

value of zJ t

OBDM as [91]:

, 0 ( 0)( 1) 2

0 2

zz fT TJ t f i

z z

T T OBDM TOBDM

T T

1 ( 1)

(2 ) 60 2

f iz

z z

T T OBDM Tt

T T

(2-31)

The OBDM contains all the information about transitions of given

multiplicit

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